Average Error: 0.9 → 0.3
Time: 1.0m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sqrt[3]{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right)\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right)}\right)} + \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)} + \lambda_1\]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sqrt[3]{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right)\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right)}\right)} + \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)} + \lambda_1
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1885569 = lambda1;
        double r1885570 = phi2;
        double r1885571 = cos(r1885570);
        double r1885572 = lambda2;
        double r1885573 = r1885569 - r1885572;
        double r1885574 = sin(r1885573);
        double r1885575 = r1885571 * r1885574;
        double r1885576 = phi1;
        double r1885577 = cos(r1885576);
        double r1885578 = cos(r1885573);
        double r1885579 = r1885571 * r1885578;
        double r1885580 = r1885577 + r1885579;
        double r1885581 = atan2(r1885575, r1885580);
        double r1885582 = r1885569 + r1885581;
        return r1885582;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1885583 = phi2;
        double r1885584 = cos(r1885583);
        double r1885585 = lambda1;
        double r1885586 = sin(r1885585);
        double r1885587 = lambda2;
        double r1885588 = cos(r1885587);
        double r1885589 = r1885586 * r1885588;
        double r1885590 = cos(r1885585);
        double r1885591 = sin(r1885587);
        double r1885592 = r1885590 * r1885591;
        double r1885593 = r1885589 - r1885592;
        double r1885594 = r1885584 * r1885593;
        double r1885595 = r1885584 * r1885588;
        double r1885596 = phi1;
        double r1885597 = cos(r1885596);
        double r1885598 = fma(r1885595, r1885590, r1885597);
        double r1885599 = r1885598 * r1885598;
        double r1885600 = exp(r1885598);
        double r1885601 = log(r1885600);
        double r1885602 = r1885599 * r1885601;
        double r1885603 = cbrt(r1885602);
        double r1885604 = r1885591 * r1885586;
        double r1885605 = r1885584 * r1885604;
        double r1885606 = r1885603 + r1885605;
        double r1885607 = atan2(r1885594, r1885606);
        double r1885608 = r1885607 + r1885585;
        return r1885608;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.9

    \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  2. Using strategy rm

  3. Applied sin-diff0.8

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  4. Using strategy rm

  5. Applied cos-diff0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\]

  6. Applied distribute-lft-in0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 + \color{blue}{\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\]

  7. Applied associate-+r+0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\left(\cos \phi_1 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}}\]

  8. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \lambda_1, \cos \phi_1\right)} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]
  9. Using strategy rm

  10. Applied add-cbrt-cube0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \lambda_1, \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \lambda_1, \cos \phi_1\right)\right) \cdot \mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \lambda_1, \cos \phi_1\right)}} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]
  11. Using strategy rm

  12. Applied add-log-exp0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sqrt[3]{\left(\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \lambda_1, \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \lambda_1, \cos \phi_1\right)\right) \cdot \color{blue}{\log \left(e^{\mathsf{fma}\left(\cos \lambda_2 \cdot \cos \phi_2, \cos \lambda_1, \cos \phi_1\right)}\right)}} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]

  13. Final simplification0.3

    \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sqrt[3]{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right)\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \lambda_2, \cos \lambda_1, \cos \phi_1\right)}\right)} + \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)} + \lambda_1\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))